Tight lower bound on geometric discord of bipartite states

We use singular value decomposition to derive a tight lower bound for geometric discord of arbitrary bipartite states. In a single shot this also leads to an upper bound of measurement-induced non locality which in turn yields that for Werner and isotropic states the two measures coincide. We also emphasize that our lower bound is saturated for all $2\otimes n$ states. Using this we show that both the generalized Greenberger-Horne-Zeilinger and $W$ states of $N$ qubits satisfy monogamy of geometric discord. Indeed, the same holds for all $N$-qubit pure states which are equivalent to $W$ states under stochastic local operations and classical communication. We show by giving an example that not all pure states of four or higher qubits satisfy monogamy.